Zhu algebra

In mathematics, the Zhu algebra and the closely related C₂-algebra, introduced by Yongchang Zhu in his PhD thesis, are two associative algebras canonically constructed from a given vertex operator algebra. Many important representation theoretic properties of the vertex algebra are logically related to properties of its Zhu algebra or C₂-algebra.

Definitions

Let $V=\underset{n\ge 0}{⨁}{V}_{(n)}$ be a graded vertex operator algebra with $V_{(0)}=\mathbb{ℂ}\mathbf{1}$ and let $Y(a,z)=\sum _{n\in \mathbb{\mathbb{Z}}}{a}_n{z}^{-n-1}$ be the vertex operator associated to $a\in V.$ Define $C_2(V)\subset V$to be the subspace spanned by elements of the form $a_{-2}b$ for $a,b\in V.$ An element $a\in V$ is homogeneous with $\mathrm{wt}a=n$ if $a\in V_{(n)}.$ There are two binary operations on $V$defined by$$a*b=\sum _{i\ge 0}(\genfrac{}{}{0ex}{}{\mathrm{wt}a}{i})a_{i-1}b,a\circ b=\sum _{i\ge 0}(\genfrac{}{}{0ex}{}{\mathrm{wt}a}{i})a_{i-2}b$$for homogeneous elements and extended linearly to all of $V$. Define $O(V)\subset V$to be the span of all elements $a\circ b$. The algebra $A(V):=V/O(V)$ with the binary operation induced by $*$ is an associative algebra called the Zhu algebra of $V$. The algebra $R_V:=V/C_2(V)$ with multiplication $a\cdot b=a_{-1}bmod{C}_2(V)$ is called the C₂-algebra of $V$.

Main properties

• The multiplication of the C₂-algebra is commutative and the additional binary operation $\{a,b\}=a_{0}bmod{C}_2(V)$ is a Poisson bracket on $R_V$which gives the C₂-algebra the structure of a Poisson algebra.
• (Zhu's C₂-cofiniteness condition) If $R_V$is finite dimensional then $V$ is said to be C₂-cofinite. There are two main representation theoretic properties related to C₂-cofiniteness. A vertex operator algebra $V$ is rational if the category of admissible modules is semisimple and there are only finitely many irreducibles. It was conjectured that rationality is equivalent to C₂-cofiniteness and a stronger condition regularity, however this was disproved in 2007 by Adamovic and Milas who showed that the triplet vertex operator algebra is C₂-cofinite but not rational. Various weaker versions of this conjecture are known, including that regularity implies C₂-cofiniteness and that for C₂-cofinite $V$ the conditions of rationality and regularity are equivalent. This conjecture is a vertex algebras analogue of Cartan's criterion for semisimplicity in the theory of Lie algebras because it relates a structural property of the algebra to the semisimplicity of its representation category.
• The grading on $V$ induces a filtration $A(V)=\bigcup _{p\ge 0}{A}_p(V)$ where $A_p(V)=\mathrm{im}(\underset{j=0}{\overset{p}{\oplus }}{V}_p\to A(V))$so that $A_p(V)\ast A_q(V)\subset A_{p+q}(V).$ There is a surjective morphism of Poisson algebras $R_V\to \mathrm{gr}(A(V))$.

Associated variety

Because the C₂-algebra $R_V$ is a commutative algebra it may be studied using the language of algebraic geometry. The associated scheme ${\tilde{X}}_V$ and associated variety $X_V$ of $V$ are defined to be $${\tilde{X}}_V:=\mathrm{Spec}(R_V),X_V:=({\tilde{X}}_V{)}_{\mathrm{r}\mathrm{e}\mathrm{d}}$$which are an affine scheme an affine algebraic variety respectively. Moreover, since $L(-1)$ acts as a derivation on $R_V$ there is an action of ${\mathbb{ℂ}}^{\ast }$ on the associated scheme making ${\tilde{X}}_V$ a conical Poisson scheme and $X_V$ a conical Poisson variety. In this language, C₂-cofiniteness is equivalent to the property that $X_V$ is a point. Example: If $W^k(\widehat{\mathfrak{𝔤}},f)$ is the affine W-algebra associated to affine Lie algebra $\widehat{\mathfrak{𝔤}}$ at level $k$ and nilpotent element $f$ then ${\tilde{X}}_{W^k(\widehat{\mathfrak{𝔤}},f)}={{𝒮}}_f$is the Slodowy slice through $f$.